The following problem is studied: find conditions on sequences $\{\gamma(r)\}$, $\{n_j\}$, and $\{v_j\}$ under which, for any sequence$\{b_k\}$ such that $\sum_{k=r}^{\infty}|b_k-b_{k+1}|\leq\gamma(r)$, $b_k\to 0$, the integral $\int_0^\pi U^p(x)/{x^q}dx$ is convergent, where $p>0$, $q\in[1-p;1)$, and $U(x):=\sum_{j=1}^{\infty}\left|\sum_{k=n_j}^{v_j}b_k \sin kx\right|$. In the case $\gamma(r)={B}/{r}$, $B>0$, this problem was studied and solved by S.A.Telyakovskii. In the case where $p\ge 1$, $q=0$, $v_j=n_{j+1}-1$, and the sequence $\{b_k\}$ is monotone, A.S.Belov obtained a criterion for the belonging of the function $U(x)$ to the space$L_p$. In Theorem1 of the present paper, we give sufficient conditions for the convergence of the above integral, which for $\gamma(r)= B/{r}$, $B>0$, coincide with Telyakovskii's sufficient conditions. In the case $\gamma(r)= O(1/{r})$, Telyakovskii's conditions may be violated, but the application of Theorem1 guarantees the convergence of the integral. The corresponding examples are given in the last section of the paper. The question on necessary conditions for the convergence of the integral $\int_0^\pi U^p(x)/{x^q}dx$, where $p>0$ and $q\in[1-p;1)$, remains open.
Keywords: trigonometric series, sums of moduli of blocks, power weight.
The paper was received by the Editorial Office on May 15, 2017.
Viktor Petrovich Zastavnyi, Dr. Phys.-Math. Sci., Prof., Donetsk National University, Universitetskaya str. 24, Donetsk, 83001, Ukraine, e-mail: zastavn@rambler.ru
Antonina Sergeevna Levadnaya, doctoral student, Donetsk National University, Universitetskaya str. 24, Donetsk, 83001, Ukraine, e-mail: last.dris@mail.ru
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